Subgroups — #30615342

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461750	C₂	1	2	yes	maximal, complemented
2	#21461738	C₂	1	2	yes	maximal, complemented
3	#21461729	C₂	1	2	yes	maximal
4	#21461703	C₂	1	1	yes	maximal, characteristic
5	#21461701	C₂	1	1	yes	maximal, characteristic
6	#21461621	C₂	1	2	yes	maximal, complemented
7	#21461575	C₂	1	2	yes	maximal
8	#21461541	C₂	1	2	yes	maximal, complemented
9	#21454241	C₂	1	1	yes	maximal, characteristic
10	D₆₁⋊#108	C₂²	1	2	yes	—
11	#16840772	C₂²	1	2	yes	—
12	#16840748	C₂²	1	2	yes	—
13	#16840741	C₂²	1	2	yes	—
14	#16840725	C₂²	1	4	yes	—
15	#16840644	C₂²	1	4	yes	—
16	#16840633	C₂²	1	4	yes	—
17	#16840631	C₂²	1	2	yes	—
18	#16840630	C₂²	1	2	yes	—
19	#16840629	C₂²	1	4	yes	—
20	C₂×#14519785	C₂²	1	1	yes	characteristic
21	C₂×#14519783	C₂²	1	1	yes	characteristic
22	#16840230	C₂²	1	1	yes	characteristic
23	#16839719	C₄	1	1	yes	characteristic, complemented
24	#16839719	C₂²	1	1	yes	characteristic
25	#16839718	C₄	1	1	yes	characteristic, complemented
26	#16839716	C₄	1	1	yes	characteristic, complemented
27	#16839716	C₂²	1	1	yes	characteristic
28	#16839714	C₄	1	1	yes	characteristic, complemented
29	#16839714	C₂²	1	1	yes	characteristic
30	M₁₆×Dic₆₁	C₄	1	1	yes	characteristic, complemented
31	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic, complemented
32	C₄×C₈×D₆₁	C₄	1	1	yes	characteristic, complemented
33	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
34	C₆₁×#405	C₄	1	1	yes	characteristic, complemented
35	D₆₁⋊M₁₆	C₂³	1	4	yes	—
36	#14519856	C₂³	1	4	yes	—
37	#14519848	—	2	2	no	—
38	#14519847	—	2	2	no	—
39	#14519846	—	2	2	no	—
40	#14519845	—	2	2	no	—
41	#14519838	—	2	2	no	—
42	#14519837	—	2	2	no	—
43	#14519836	—	2	2	no	—
44	#14519835	—	2	2	no	—
45	#14519831	—	2	4	no	—
46	C₈×#6923	—	2	4	no	—
47	C₂×#11758222	C₂³	1	1	yes	characteristic
48	#14519785	—	2	2	no	—
49	#14519785	—	2	2	no	—
50	#14519783	—	2	2	no	—
51	#14519783	—	2	2	no	—
52	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
53	C₂×#11758189	C₂×C₄	1	1	yes	characteristic
54	C₂×#11758189	C₂×C₄	1	1	yes	characteristic
55	C₂×#11758189	C₂³	1	1	yes	characteristic
56	C₂×#11758189	C₂³	1	1	yes	characteristic
57	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
62	#14519742	—	2	1	no	—
63	#14519739	—	2	1	no	—
64	#14519737	—	2	1	no	—
65	#14519737	—	2	1	no	—
66	#14519734	C₂×C₄	1	1	yes	characteristic
67	#14519734	C₂³	1	1	yes	characteristic
68	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
69	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
70	C₄²×D₆₁	C₂³	1	1	yes	characteristic
71	#14519688	C₂×C₄	1	1	yes	characteristic
72	#14519688	C₂×C₄	1	1	yes	characteristic
73	#14519687	C₂×C₄	1	1	yes	characteristic
74	#14519687	C₂×C₄	1	1	yes	characteristic
75	C₄×#92418	C₂×C₄	1	1	yes	characteristic
76	C₄×#92418	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
83	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
85	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
86	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
87	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₆₁	#41	1	1	yes	characteristic
92	C₂×C₂²⋊D₆₁	#41	1	1	yes	characteristic
93	C₂²×Dic₆₁	—	2	1	no	—
94	#11758222	—	2	2	no	—
95	#11758222	—	2	2	no	—
96	C₂×D₂₄₄	#41	1	1	yes	characteristic
97	C₂×C₄×D₆₁	—	2	1	no	—
98	C₂×C₄×D₆₁	—	2	1	no	—
99	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₆₁	C₂⁴	1	1	yes	characteristic
101	C₂×Dic₁₂₂	#41	1	1	yes	characteristic
102	C₂²⋊#6923	—	2	4	no	—
103	#11758217	—	2	4	no	—
104	C₄⋊#6923	—	2	4	no	—
105	C₄×#6923	—	2	4	no	—
106	#11758213	—	2	4	no	—
107	D₆₁⋊C₈	—	2	4	no	—
108	M₁₆×C₆₁	—	2	2	no	—
109	C₂×C₈×C₆₁	—	2	1	no	—
110	C₂×C₈×C₆₁	—	2	1	no	—
111	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₆₁	—	2	1	no	—
116	C₂²⋊C₄×C₆₁	—	2	1	no	—
117	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
119	#11758198	—	2	1	no	—
120	#11758198	—	2	1	no	—
121	#11758196	—	2	1	no	—
122	#11758196	—	2	1	no	—
123	C₄×Dic₆₁	—	2	1	no	—
124	C₄×Dic₆₁	—	2	1	no	—
125	C₄×Dic₆₁	—	2	1	no	—
126	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
128	#11758194	—	2	2	no	—
129	C₂×#92418	—	2	1	no	—
130	C₂×#92418	—	2	1	no	—
131	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
132	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
133	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
134	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
135	#11758189	—	2	2	no	—
136	#11758189	—	2	2	no	—
137	#11758189	—	2	2	no	—
138	#11758189	—	2	2	no	—
139	C₈×D₆₁	—	2	1	no	—
140	C₈×D₆₁	—	2	1	no	—
141	C₈×D₆₁	—	2	1	no	—
142	C₈×D₆₁	—	2	1	no	—
143	C₈×D₆₁	#41	1	2	yes	—
144	C₈×D₆₁	#41	1	2	yes	—
145	C₈×D₆₁	#41	1	2	yes	—
146	C₈×D₆₁	#41	1	2	yes	—
147	C₂³×C₆₁	#117	1	1	yes	characteristic
148	C₂²×D₆₁	#117	1	1	yes	characteristic
149	C₂²×D₆₁	C₂×#41	1	1	yes	characteristic
150	C₂×#6923	—	4	4	no	—
151	C₂×C₄×C₆₁	—	2	1	no	—
152	C₂×C₄×C₆₁	—	2	1	no	—
153	C₂×C₄×C₆₁	—	2	1	no	—
154	C₂×C₄×C₆₁	—	2	1	no	—
155	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
156	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
157	C₂²⋊D₆₁	—	2	2	no	—
158	C₂²⋊D₆₁	—	2	2	no	—
159	C₂×Dic₆₁	—	2	1	no	—
160	C₂×Dic₆₁	—	2	1	no	—
161	C₂×Dic₆₁	—	4	2	no	—
162	C₂×Dic₆₁	#117	1	1	yes	characteristic
163	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
164	D₂₄₄	—	2	2	no	—
165	C₄×D₆₁	—	2	1	no	—
166	C₄×D₆₁	—	2	1	no	—
167	C₄×D₆₁	—	4	2	no	—
168	C₄×D₆₁	—	4	2	no	—
169	C₄×D₆₁	C₂×#41	1	2	yes	—
170	C₄×D₆₁	C₂×#41	1	2	yes	—
171	Dic₁₂₂	—	2	2	no	—
172	#92420	—	4	4	no	—
173	C₈×C₆₁	—	2	1	no	—
174	C₈×C₆₁	—	2	1	no	—
175	C₈×C₆₁	#117	1	1	yes	characteristic
176	C₈×C₆₁	#117	1	1	yes	characteristic
177	C₈×C₆₁	#117	1	2	yes	—
178	#92418	—	2	1	no	—
179	#92418	—	2	1	no	—
180	#92418	#117	1	1	yes	characteristic
181	#92418	#117	1	1	yes	characteristic
182	#92418	#117	1	2	yes	—
183	#20787	—	61	1	no	maximal, Sylow
184	C₂²×C₆₁	—	2	1	no	—
185	C₂²×C₆₁	—	4	1	no	—
186	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
187	C₂×D₆₁	—	2	1	no	—
188	C₂×D₆₁	—	4	1	no	—
189	C₂×D₆₁	—	4	1	no	—
190	C₂×D₆₁	#529	1	2	yes	—
191	#6923	—	8	4	no	—
192	C₄×C₆₁	—	2	1	no	—
193	C₄×C₆₁	—	4	1	no	—
194	C₄×C₆₁	—	4	1	no	—
195	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
196	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
197	Dic₆₁	—	2	2	no	—
198	Dic₆₁	—	2	1	no	—
199	Dic₆₁	—	4	1	no	—
200	Dic₆₁	#529	1	2	yes	—
201	C₄×#131	—	61	1	no	—
202	#1841	—	61	2	no	—
203	#1833	—	61	2	no	—
204	#1827	—	61	2	no	—
205	#1813	—	61	1	no	—
206	#1812	—	61	1	no	—
207	#1772	—	61	2	no	—
208	#1757	—	61	2	no	—
209	#1747	—	61	2	no	—
210	C₂×C₆₁	—	2	1	no	—
211	C₂×C₆₁	—	4	1	no	—
212	C₂×C₆₁	#2283	1	1	yes	characteristic
213	D₆₁	—	2	2	no	—
214	D₆₁	—	4	1	no	—
215	C₂×#131	—	61	1	no	—
216	C₂×#131	—	61	1	no	—
217	C₄×#41	—	61	1	no	—
218	C₂×#108	—	61	2	no	—
219	#427	—	61	2	no	—
220	#413	—	61	2	no	—
221	#410	—	61	2	no	—
222	#405	—	61	1	no	—
223	#405	—	61	1	no	—
224	#405	—	61	1	no	—
225	#405	—	61	1	no	—
226	#405	—	61	1	no	—
227	#405	—	61	4	no	—
228	#405	#6923	1	1	yes	characteristic, complemented
229	C₄×M₁₆	—	61	1	no	—
230	C₄×M₁₆	—	61	1	no	—
231	C₄×M₁₆	—	61	1	no	—
232	C₂×C₄×C₈	—	61	1	no	—
233	C₂×C₄×C₈	—	61	1	no	—
234	C₂×C₄×C₈	—	61	1	no	—
235	#344	—	61	4	no	—
236	#339	—	61	4	no	—
237	#338	—	61	2	no	—
238	H(3,ℤ/4)	—	61	2	no	—
239	H(3,ℤ/4)	—	61	4	no	—
240	C₆₁	#20787	1	1	yes	characteristic, complemented, Sylow
241	C₂×#41	—	61	1	no	—
242	#131	—	122	2	no	—
243	#131	—	122	2	no	—
244	#131	—	122	2	no	—
245	#131	—	122	2	no	—
246	C₂×M₁₆	—	61	1	no	—
247	C₂×M₁₆	—	61	1	no	—
248	C₂×M₁₆	—	61	1	no	—
249	C₂×M₁₆	—	61	1	no	—
250	C₂×M₁₆	—	61	1	no	—
251	C₂×M₁₆	—	61	4	no	—
252	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
253	C₂²×C₈	—	61	1	no	—
254	C₂²×C₈	—	61	1	no	—
255	C₂²×C₈	—	61	1	no	—
256	C₂²×C₈	—	61	1	no	—
257	C₂²×C₈	—	61	1	no	—
258	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
259	C₄×Q₈	—	122	1	no	—
260	C₄×D₄	—	122	1	no	—
261	C₄×D₄	—	122	1	no	—
262	C₄×D₄	—	122	1	no	—
263	#117	—	61	1	no	—
264	#117	—	61	1	no	—
265	#117	—	61	4	no	—
266	#117	C₂×#6923	1	1	yes	characteristic
267	C₂×C₄²	—	61	1	no	—
268	C₂×C₄²	—	61	1	no	—
269	C₂×C₄²	—	61	1	no	—
270	#108	—	122	2	no	—
271	#108	—	122	2	no	—
272	#107	—	122	2	no	—
273	#106	—	122	2	no	—
274	#101	—	122	2	no	—
275	#100	—	122	2	no	—
276	#99	—	122	2	no	—
277	#99	—	122	2	no	—
278	#97	—	61	1	no	—
279	#97	—	61	1	no	—
280	#97	—	61	1	no	—
281	#97	—	61	1	no	—
282	#97	—	122	4	no	—
283	#97	C₂×#6923	1	1	yes	characteristic
284	#97	C₂×#6923	1	1	yes	characteristic
285	C₄×C₈	—	61	1	no	—
286	C₄×C₈	—	61	1	no	—
287	C₄×C₈	—	61	1	no	—
288	C₄×C₈	—	61	1	no	—
289	C₄×C₈	—	61	1	no	—
290	C₄×C₈	—	61	1	no	—
291	C₄×C₈	—	61	1	no	—
292	C₄×C₈	—	61	1	no	—
293	C₄×C₈	—	122	4	no	—
294	C₄×C₈	C₂×#6923	1	1	yes	characteristic
295	C₄×C₈	C₂×#6923	1	1	yes	characteristic
296	#41	—	122	2	no	—
297	#41	—	122	2	no	—
298	C₂×Q₈	—	61	1	no	—
299	C₂×D₄	—	61	1	no	—
300	C₂×D₄	—	61	1	no	—
301	C₂×D₄	—	61	1	no	—
302	C₂²×C₄	—	61	1	no	—
303	C₂²×C₄	—	61	1	no	—
304	C₂²×C₄	—	122	1	no	—
305	C₂²×C₄	—	122	1	no	—
306	C₂²×C₄	—	122	1	no	—
307	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
308	M₁₆	—	2	2	no	—
309	M₁₆	—	122	2	no	—
310	M₁₆	—	122	2	no	—
311	M₁₆	—	122	2	no	—
312	M₁₆	—	122	2	no	—
313	M₁₆	—	122	2	no	—
314	M₁₆	—	122	4	no	—
315	M₁₆	—	122	4	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	61	1	no	—
319	C₂×C₈	—	61	1	no	—
320	C₂×C₈	—	61	2	no	—
321	C₂×C₈	—	61	2	no	—
322	C₂×C₈	—	61	2	no	—
323	C₂×C₈	—	61	2	no	—
324	C₂×C₈	—	61	1	no	—
325	C₂×C₈	—	61	1	no	—
326	C₂×C₈	—	122	1	no	—
327	C₂×C₈	—	122	1	no	—
328	C₂×C₈	—	122	1	no	—
329	C₂×C₈	—	122	1	no	—
330	C₂×C₈	—	122	1	no	—
331	C₂×C₈	—	122	1	no	—
332	C₂×C₈	—	122	4	no	—
333	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
334	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
335	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
336	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	122	1	no	—
339	C₄⋊C₄	—	122	1	no	—
340	C₄⋊C₄	—	122	4	no	—
341	C₂²⋊C₄	—	2	1	no	—
342	C₂²⋊C₄	—	122	1	no	—
343	C₂²⋊C₄	—	122	1	no	—
344	C₂²⋊C₄	—	122	4	no	—
345	C₄²	—	61	1	no	—
346	C₄²	—	61	1	no	—
347	C₄²	—	122	1	no	—
348	C₄²	—	122	1	no	—
349	C₄²	—	122	4	no	—
350	C₄²	—	122	1	no	—
351	C₄²	C₂²×#6923	1	1	yes	characteristic
352	C₄²	C₂²×#6923	1	1	yes	characteristic
353	C₂³	—	61	1	no	—
354	C₂³	—	61	1	no	—
355	C₂³	#14519856	1	1	yes	characteristic
356	Q₈	—	122	2	no	—
357	D₄	—	122	2	no	—
358	D₄	—	122	2	no	—
359	D₄	—	122	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	61	1	no	—
365	C₂×C₄	—	61	2	no	—
366	C₂×C₄	—	61	2	no	—
367	C₂×C₄	—	61	1	no	—
368	C₂×C₄	—	122	1	no	—
369	C₂×C₄	—	122	1	no	—
370	C₂×C₄	—	122	1	no	—
371	C₂×C₄	—	122	1	no	—
372	C₂×C₄	—	244	2	no	—
373	C₂×C₄	—	244	2	no	—
374	C₂×C₄	—	244	2	no	—
375	C₂×C₄	—	244	4	no	—
376	C₂×C₄	#14519856	1	1	yes	characteristic
377	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	—	61	2	no	—
381	C₈	—	61	1	no	—
382	C₈	—	61	1	no	—
383	C₈	—	122	1	no	—
384	C₈	—	122	1	no	—
385	C₈	—	244	4	no	—
386	C₈	#14519856	1	1	yes	characteristic
387	C₈	#14519856	1	1	yes	characteristic
388	C₈	#14519856	1	2	yes	—
389	C₂²	—	2	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	61	2	no	—
392	C₂²	—	122	1	no	—
393	C₂²	—	244	1	no	—
394	C₂²	—	244	1	no	—
395	C₂²	C₂×#14519856	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	4	1	no	—
398	C₄	—	4	1	no	—
399	C₄	—	61	2	no	—
400	C₄	—	122	2	no	—
401	C₄	—	122	1	no	—
402	C₄	—	488	4	no	—
403	C₄	—	244	1	no	—
404	C₄	C₂×#14519856	1	1	yes	characteristic
405	C₄	C₂×#14519856	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	4	1	no	—
408	C₂	—	122	2	no	—
409	C₂	—	244	1	no	—
410	C₂	#21462266	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14519856
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#405	C₄
socle	C₂×C₆₁	#2283
prime radical	C₂×#11758222	C₂³
cosocle	C₂×C₄×D₆₁	C₂⁴
hypercenter	#405	#6923
hypocenter	C₆₁	#20787
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #97 ◁ #405

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #21461750	C₂	semidirect
2. #21461738	C₂	semidirect
6. #21461621	C₂	semidirect
8. #21461541	C₂	semidirect
23. #16839719	C₄	semidirect
25. #16839718	C₄	semidirect
26. #16839716	C₄	semidirect
28. #16839714	C₄	semidirect
30. M₁₆×Dic₆₁	C₄	semidirect
31. C₂×C₈×Dic₆₁	C₄	semidirect
32. C₄×C₈×D₆₁	C₄	semidirect
34. C₆₁×#405	C₄	semidirect
228. #405	#6923	semidirect
240. C₆₁	#20787	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21461750	2
#21461738	2
#21461729	2
#21461703	2
#21461701	2
#21461621	2
#21461575	2
#21461541	2
#21454241	2
#20787	61